Further insight into the ability of the ML-RDGF model to predict the velocity gradient dynamics can be obtained by considering its predictions for the velocity gradient invariants \(Q\equiv -{a}{:a}/2\)  and \(R\equiv -({a \cdot } {a}) {:a}/3\) . The invariant \(Q\)  measures the relative strength of the local strain rate and vorticity in the flow, with \(Q>0\)  denoting vorticity-dominated regions of the flow, while \(R\)  measures the relative importance of the local strain self-amplification and enstrophy production, with \(R<0\)  denoting regions dominated by enstrophy production [1]}. The joint PDFs of \(Q\)  and \(R\)  from the ML-RDGF model for M1, M2, M3 are presented in Figure REF , along with the DNS results for comparison. As observed previously [2]}, the ML-RDGF captures the main features of the \(Q,R\)  joint-PDF well, including the signature sheared-drop shape of the joint-PDF, which is preserved by the model as \(_\lambda \)  is increased. The joint-PDF contours extend along the right Viellefosse tail, and the PDF is concentrated in the quadrants \(Q>0, R<0\)  and \(Q<0, R>0\) , consistent with previous studies [3]}, [2]}. For M1, the joint-PDF is more compact compared to the DNS, indicating that the model underpredicts the probability of large values of \(Q,R\) . The agreement between the ML-RDGF model and DNS data was shown to be better in [2]}, however, their DNS data was for \(_\lambda =430\) , indicating that the quantitive accuracy of the model is better at higher \(_\lambda \) . Considering the results for M2 and M3 shows that as \(_\lambda \)  increases, the contours of the joint-PDF spread out in the \(Q,R\)  phase-space, showing that the model captures the effect of \(_\lambda \)  on the intermittency in the flow.
